Free access
Issue
ESAIM: M2AN
Volume 41, Number 1, January-February 2007
Page(s) 169 - 185
DOI http://dx.doi.org/10.1051/m2an:2007011
Published online 26 April 2007
  1. F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Birkhäuser (2004).
  2. A. Bressan, H.K. Jenssen and P. Baiti, An instability of the Godunov Scheme. arXiv:math.AP/0502125 v2 (2005).
  3. M.J. Castro, J. Macías and C. Parés, A Q-Scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. Math. Mod. Num. Anal. 35 (2001) 107–127. [CrossRef] [EDP Sciences]
  4. F. Coquel, D. Diehl, C. Merkle and C. Rohde, Sharp and diffuse interface methods for phase transition problems in liquid-vapour flows, in Numerical Methods for Hyperbolic and Kinetic Problems, IRMA Lectures in Mathematics and Theoretical Physics, Proceedings of CEMRACS 2003.
  5. G. Dal Maso, P.G. LeFloch and F. Murat, Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483–548. [MathSciNet]
  6. F. De Vuyst, Schémas non-conservatifs et schémas cinétiques pour la simulation numérique d'écoulements hypersoniques non visqueux en déséquilibre thermochimique. Ph.D. thesis, University of Paris VI, France (1994).
  7. E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer (1996).
  8. L. Gosse, A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput. Math. Appl. 39 (2000) 135–159. [CrossRef] [MathSciNet]
  9. L. Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. Math. Models Methods Appl. Sci. 11 (2001) 339–365. [CrossRef] [MathSciNet]
  10. J.M. Greenberg and A.Y. LeRoux, A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1–16. [CrossRef] [MathSciNet]
  11. J.M. Greenberg, A.Y. LeRoux, R. Baraille and A. Noussair, Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34 (1997) 1980–2007. [CrossRef] [MathSciNet]
  12. A. Harten, P.D. Lax and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35–61. [CrossRef] [MathSciNet]
  13. T. Hou and P.G. LeFloch, Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comp. 62 (1994) 497–530. [CrossRef] [MathSciNet]
  14. E. Isaacson and B. Temple, Convergence of the 2 x 2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55 (1995) 625–640. [CrossRef] [MathSciNet]
  15. P.D. Lax and B. Wendroff, Systems of conservation laws. Comm. Pure Appl. Math. 13 (1960) 217–237. [CrossRef] [MathSciNet]
  16. P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form. Institute Math. Appl., Minneapolis, Preprint 593 (1989).
  17. P.G. LeFloch, Graph solutions of nonlinear hyperbolic systems. J. Hyper. Differ. Equa. 2 (2004) 643–689.
  18. P.G. LeFloch and A.E. Tzavaras, Representation of weak limits and definition of nonconservative products. SIAM J. Math. Anal. 30 (1999) 1309–1342. [CrossRef] [MathSciNet]
  19. R.J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comp. Phys. 146 (1998) 346–365. [NASA ADS] [CrossRef] [MathSciNet]
  20. C. Parés, Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44 (2006) 300–321. [CrossRef] [MathSciNet]
  21. C. Parés and M. Castro, On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow water systems. Math. Mod. Num. Anal. 38 (2004) 821–852. [CrossRef] [EDP Sciences]
  22. A.I. Volpert, The space BV and quasilinear equations. Math. USSR Sbornik 73 (1967) 225–267. [CrossRef]

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